Assumption 1(b) (Quadratic Growth (QG) condition [1]})  
The function family 
\(\mathcal {F} = \lbrace  f_{ : \mathbb {R}^m \rbrace  is assumed to satisfy the QG condition with parameter C^*_{n,\delta }, if with probability \ge 1 - \delta  over the dataset D_n,{\begin{@align}{1}{-1} \forall \mathbb {R}^m,\ \mathrm {Err}_{n} (*)\ge C^*_{n,\delta } \Vert   * \Vert _2^2.\end{@align}}}\begin{remark} \cite [Theorem 2]{doi:10.1137/S1052623499359178}Restricting to functions f which have L-Lipschitz continuous gradients, if f \in \mathrm {RSI} (C), then it implies that f \in \mathrm {QG} (2C/L). This implies that up to the value of the parameter, the QG condition is weaker than RSI, under a smoothness constraint on the considered functions.\end{remark}\) Note that under the QG condition, gradients of \(\mathrm {Err}_{n} (*)\)  are no longer constrained to be positively correlated with the \( *\) . In fact, the absence of this feature proves to be a significant barrier for first-order methods to generalize well. To circumvent this issue, we motivate and introduce the notion of approximate first-order stationary interpolators in the next section, and characterize their statistical performance. This imposes a stronger requirement on the algorithm than just returning an arbitrary stationary point of the training objective \(\mathcal {L} ( + R (\) .
